335 research outputs found
An adaptive discontinuous finite volume method for elliptic problems
AbstractAn adaptive discontinuous finite volume method is developed and analyzed in this paper. We prove that the adaptive procedure achieves guaranteed error reduction in a mesh-dependent energy norm and has a linear convergence rate. Numerical results are also presented to illustrate the theoretical analysis
Higher order weakly over-penalized symmetric interior penalty methods
In this paper we study higher order weakly over-penalized symmetric interior penalty methods for second-order elliptic boundary value problems in two dimensions. We derive hp error estimates in both the energy norm and the norm and present numerical results that corroborate the theoretical results. © 2012 Elsevier B.V. All rights reserved. L
Scaling-robust built-in a posteriori error estimation for discontinuous least-squares finite element methods
A convincing feature of least-squares finite element methods is the built-in
a posteriori error estimator for any conforming discretization. In order to
generalize this property to discontinuous finite element ansatz functions, this
paper introduces a least-squares principle on piecewise Sobolev functions for
the solution of the Poisson model problem in 2D with mixed boundary conditions.
It allows for fairly general discretizations including standard piecewise
polynomial ansatz spaces on triangular and polygonal meshes. The presented
scheme enforces the interelement continuity of the piecewise polynomials by
additional least-squares residuals. A side condition on the normal jumps of the
flux variable requires a vanishing integral mean and enables a natural
weighting of the jump in the least-squares functional in terms of the mesh
size. This avoids over-penalization with additional regularity assumptions on
the exact solution as usually present in the literature on discontinuous LSFEM.
The proof of the built-in a posteriori error estimation for the over-penalized
scheme is presented as well. All results in this paper are robust with respect
to the size of the domain guaranteed by a suitable weighting of the residuals
in the least-squares functional. Numerical experiments exhibit optimal
convergence rates of the adaptive mesh-refining algorithm for various
polynomial degrees
A Computational Study of the Weak Galerkin Method for Second-Order Elliptic Equations
The weak Galerkin finite element method is a novel numerical method that was
first proposed and analyzed by Wang and Ye for general second order elliptic
problems on triangular meshes. The goal of this paper is to conduct a
computational investigation for the weak Galerkin method for various model
problems with more general finite element partitions. The numerical results
confirm the theory established by Wang and Ye. The results also indicate that
the weak Galerkin method is efficient, robust, and reliable in scientific
computing.Comment: 19 page
Weighted Error Estimates for Transient Transport Problems Discretized Using Continuous Finite Elements with Interior Penalty Stabilization on the Gradient Jumps
In this paper we consider the semi-discretization in space of a first order scalar transport equation. For the space discretization we use standard continuous finite elements with a stabilization consisting of a penalty on the jump of the gradient over element faces. We recall some global error estimates for smooth and rough solutions and then prove a new local error estimate for the transient linear transport equation. In particular we show that for the stabilized method the effect of non-smooth features in the solution decay exponentially from the space time zone where the solution is rough so that smooth features will be transported unperturbed. Locally the L2-norm error converges with the expected order O(hk+12), if the exact solution is locally smooth. We then illustrate the results numerically. In particular we show the good local accuracy in the smooth zone of the stabilized method and that the standard Galerkin fails to approximate a solution that is smooth at the final time if underresolved features have been present in the solution at some time during the evolution
Analysis of discontinuous Galerkin methods using mesh-dependent norms and applications to problems with rough data
We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine some problems with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control
Exponentially Fitted Discontinuous Galerkin Schemes for Singularly Perturbed Problems
New discontinuous Galerkin schemes in mixed form are introduced for symmetric elliptic problems of second order. They exhibit reduced connectivity with respect to the standard ones. The modifications in the choice of the approximation spaces and in the stabilization term do not spoil the error estimates. These methods are then used for designing new exponentially fitted schemes for advection dominated equations. The presented numerical tests show the good performances of the proposed schemes.Fil: Lombardi, Ariel Luis. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; ArgentinaFil: Pietra, Paola. Consiglio Nazionale delle Ricerche. Istituto di Matematica Applicata e Tecnologie Informatiche; Itali
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